I understand all of the cases for the option bounds (including the adjustment to the minimum value of the American call option). The only item that I am having a doubt about is the upper bound to the European call option. Max value of European call option <= S0 The max value of the American call option is also <= S0 (CFAI v6 p102). Since American options are worth at least as much as European options, why isn’t there an adjustment downward for the max value of the European call? Something like <= S0e^-rt? It is 12:00am so I hope I’m missing something simple. I’ve looked around the forum and have not found a discussion on this item specifically. Any pointers are welcome. thanks
Unlike put options, the potential payoff for the holder of calls (european or american) is unlimited. As such it would not make sense to have different upper bounds. The max value is therefore S0.
The possible unlimited payoff doesn’t have much to do with it. First, remember that American calls and European calls on stocks are pretty much worth the same because early exercise of a call option is only a good idea in pretty odd situations (basically when the time value of the earnings from exercise exceeds the time value of the option which only happens with low vol and high interest rates). That means you should think that since their values are about the same, their max value might be something like the same. Next, the reason that a call can be worth almost the same as its underlying asset is that it is certain to be exercised and the strike price is 0. That’s a ridiculous option (unless there’s some tax ploy or similar there which is definitely possible) but in that case, early exercise is just not part of the picture. If you want a really high delta call, you almost certainly don’t need a strike price near 0. People get too hung up on these American vs European option distinction. The differences are subtle and the big picture is much more important.
hey JoeyDVivre. then could you also please explain why upper bounds on the American / European put options are different. As I understand early put exercise also makes sense only when interest rates are high. Thanks
Not in exactly the same way as a call (although maybe they are the same way in a very broad sense). (Forget about taxes, etc.) If a stock goes to 0, you are going to exercise the put because there is no more money to be made by holding the put. Even if interest rates are 0, you would rather have $X cash than a put option worth $X because the cash is more fungible. They just don’t take put options as payment on your Amex bill. An American put has a bound of X because the stock might go to 0 instantly. The European put has the bound of X/(1+r)^t because it might go to 0 instantly but you’re not getting paid until the option expires.
thanks for your response. but a stock going to 0 is not a very likely situation either, perhaps even less likely than low volatility / high rate / dividend payout scenarios that could lead to early call exercise. so if, as you reason, american / european calls are worth about the same (and have equal upper bounds) because early exercise of a call option is only a good idea in pretty odd situations, shouldnt the same reasoning apply for put options also? in fact if interest rate levels are assumed to be low under normal situations, X would be approximately equal to X/(1+r)^t and the put upper bounds would be equal. but the theoretical formula still allows for different rate scenarios by having “r” in the denominator. i would still think the reason why european / american calls have the same theoretical upper bound is not because early exercise of an american call could be a rare scenario. low volatility / high rates / dividend payouts are not that unlikely to be entirely discounted in a theoretical model.
The upper bounds don’t have anything to do with how rarely they are achieved. They are just bounds. The upper bounds are the same because the most they can be worth is the stock price. Think about it - what would you pay for an American call on a $50 stock with strike price of 0? It’s the same as owning the stock. A European call with a strike of 0 is worth the same amount as the stock on day T for sure and for certain and therefore it must be worth the same at all points prior to T.
The only difference between European and American options are the puts. Puts have a defined upper limit and since European puts can only be exercised at expiration, any potential payoff has to be discounted at the risk-free rate. American puts reach maximum payoff whenever the underlying hits zero anytime leading up to expiration.
yes i agree that upper bounds don’t have anything to do with how rarely they are achieved, thats exactly my point. i am just trying to figure out a response to the question initiating this thread which was - why isn’t there an adjustment downward for the max value of the European call? and i think one way to look at it would be in terms of payoff. one would never pay more than the max potential payoff, and certainly never more than the underlying asset. the call relationship therefore becomes american call <= min [S0, max payoff] european call < = min [S0, max payoff/(1+r)^t] since max payoff is unlimited, either call <= S0.
ok, a little odd but if it works for you, it works for me.
Good point! I don’t see why your not right. fullofquestions Wrote: ------------------------------------------------------- > I understand all of the cases for the option > bounds (including the adjustment to the minimum > value of the American call option). The only item > that I am having a doubt about is the upper bound > to the European call option. > > Max value of European call option <= S0 > > The max value of the American call option is also > <= S0 (CFAI v6 p102). Since American options are > worth at least as much as European options, why > isn’t there an adjustment downward for the max > value of the European call? Something like <= > S0e^-rt? > > It is 12:00am so I hope I’m missing something > simple. I’ve looked around the forum and have not > found a discussion on this item specifically. Any > pointers are welcome. > > thanks
This is a classic case of “sandwiching”. A european call is worth not more than an american call and may be worth less. Suppose that it is worth less than the american call. If we let the strike price go to 0, the european call value goes to the asset price. An american call can’t be worth more than the asset price (o/w I would short the call, buy the asset and happily deliver the asset when the call is exercised and pocket the difference). That means that the american call value is “sandwiched” between the asset price bound and the european option price. Mathematicians use sandwiching all the time to find limits of functions.
fullofquestions Wrote: ------------------------------------------------------- The only item > that I am having a doubt about is the upper bound > to the European call option. >why > isn’t there an adjustment downward for the max > value of the European call? Reason that there is no adjustment for American option is that American option’s value is sometime equal European option, sometime not. See otherwise also (it is for level II, however it is relevant for this question) http://www.analystforum.com/phorums/read.php?12,1140588,1141079#msg-1141079
“http://www.analystforum.com/phorums/read.php?12,1140588,1141079#msg-1141079” is one confusing thread with a bunch of errors in it. For example, for “For Call & Put Options on FUTURES: American > European IF DEEP IN THE MONEY but if on or out of the money, American = European?” and you answered “Correct” which stood. But that’s ignoring an important piece of the time value of the option. If an American option under some circumstances should be exercised early, then the option is worth more now regardless of whether it should be exercised now. That’s because there is a possibility that you can make more money from the American option than the European option, so the American option ought to be worth P(more money)*more money + European option (and for all you quant types, you can throw an integral in there). Also the last query about American vs European options on forwards is for the reason given as a question in the last query. That was a direct question worth 3 pts on the 2004 Level II exam and was the problem assigned to me to grade that year. Edit: Extra word in there
JoeyDVivre Wrote: > For example, for “For Call & Put Options on > FUTURES: American > European IF DEEP IN THE MONEY > but if on or out of the money, American = > European?” > > and you answered “Correct” which stood. > > But that’s ignoring an important piece of the time > value of the option. If an American option under > some circumstances should be exercised early, then > the option is worth more now regardless of whether > it should be exercised now. That’s because there > is a possibility that you can make more money from > the American option than the European option, so > the American option ought to be worth P(more > money)*more money + European option (and for all > you quant types, you can throw an integral in > there). > > Also the last query about American vs European > options on forwards is for the reason given as a > question in the last query. That was a direct > question worth 3 pts on the 2004 Level II exam and > was the problem assigned to me to grade that > year. JoeyDVivre Not sure I understand your point. Why is “FUTURES: American > European IF DEEP IN THE MONEY but if on or out of the money, American =European?” not correct? Where does it have to do with TV of money?
Not TV of money. TV of option.
JoeyDVivre Wrote: ------------------------------------------------------- > Not TV of money. TV of option. Thanks, but I am still not clear about your original posting commenting “FUTURES: American > European IF DEEP IN THE MONEY but if on or out of the money, American =European?” is not correct
Here’s a general statement: Asset: American > European if deep in the money implies American > European if the option has not expired (obviously there will be plenty of times where the difference is less than a tick so they will seem equal).
Not sure we are talking past each other. So here I what I remember (it has been a while), to make sure there is no misunderstanding: 1. Call & Put Options on FUTURES: American > European IF DEEP IN THE MONEY If deep in the money, then exercising the option NOW (American option) will give long position in the underlying futures contract PLUS a cash amount equal to the current futures price - exercise price (due to the mark-to-market feature). This access to cash is of value since you can use it to invest earning more than risk free. Therefore, American option is worth more than the European option. 2. Call & Put Options on FORWARDS: American = European since exercising the forward does not give you access to the excess cash, since there is no mark-to-market feature. I am quoting directly from Don Chance in “Analysis of derivatives for the CFA program” to support my arguments: “If the option is on a forward contract instead of a futures contract, however, these arguments are overshadowed by the fact that a forward contract does not pay off until expiration, in contrast to the mark-to-market procedure of futures contracts. Thus, if one exercised either a call or a put on a forward contract early, doing so would only establish a long or short position in a forward contract. This position would not pay any cash until expiration. No justification exists for exercising early if one cannot generate any cash from the exercise. Therefore, an American call on a forward contract is the same as a European call on a forward contract, but American calls on futures are different from European calls on futures and carry higher prices.” Let me know if it is still not clear
elcfa Wrote: ------------------------------------------------------- > Not sure we are talking past each other. > > So here I what I remember (it has been a while), > to make sure there is no misunderstanding: > > > 1. Call & Put Options on FUTURES: American > > European IF DEEP IN THE MONEY > > If deep in the money, then exercising the option > NOW (American option) will give long position in > the underlying futures contract PLUS a cash amount > equal to the current futures price - exercise > price (due to the mark-to-market feature). This > access to cash is of value since you can use it to > invest earning more than risk free. Therefore, > American option is worth more than the European > option. > True, which is why an American option will always be worth more than a European one for futures contracts (which barely matters because you trade what is available on the exchange for futures). > > 2. Call & Put Options on FORWARDS: American = > European since exercising the forward does not > give you access to the excess cash, since there is > no mark-to-market feature. > > I am quoting directly from Don Chance in “Analysis > of derivatives for the CFA program” to support my > arguments: > > “If the option is on a forward contract instead of > a futures contract, however, these > arguments are overshadowed by the fact that a > forward contract does not pay off until > expiration, in contrast to the mark-to-market > procedure of futures contracts. Thus, if one > exercised either a call or a put on a forward > contract early, doing so would only establish a > long or short position in a forward contract. This > position would not pay any cash until expiration. > No justification exists for exercising early if > one cannot generate any cash from the > exercise. Therefore, an American call on a forward > contract is the same as a European call > on a forward contract, but American calls on > futures are different from European calls on > futures and carry higher prices.” > > Let me know if it is still not clear Don Chance is right and you’re not. In the following sentence, you need to replace “exercising the forward” with “exercising the option”. BTW - There may well be a marking to market in a forward contract as a credit feature, it’s just that you can’t access this cash until the forward expires. However, if someone can’t make the mark-to-market requirement, it’s better to know sooner than later. If you trade interbank FX forwards through a broker they will sure as heck mark them every day and call you up for margin if you are losing money. “2. Call & Put Options on FORWARDS: American = European since exercising the forward does not give you access to the excess cash, since there is no mark-to-market feature.”